In the talk, I discuss previous works on the arithmetic of various twisted special $L$-values and dynamical phenomena behind them. Main emphasis will be put on the problem of estimating several exponential sums such as Kloosterman sums and its relation to the problem of non-vanishing of special $L$-values with cyclotomic twists. A distribution of homological cycles on the modular curves will also be discussed and as a consequence, some results on a conjecture of Mazur-Rubin-Stein about the distribution of period integrals of elliptic modular forms will be presented.

We show that the bipolar filtration of the smooth concordance group of topologically slice knots introduced by Cochran, Harvey and Horn has nontrivial graded quotients at every stage. To detect a nontrivial element in the quotient, the proof uses Cheeger-Gromov $L^2$ $\rho$-invariants and infinitely many Heegaard Floer correction term invariants simultaneously. This is joint work with Jae Choon Cha.

The standard problem of optical tomography is to obtain information about the optical parameters inside of an object by making optical measurements on the boundary. Acousto-optic tomography is a variation of this problem where the object is perturbed by an acoustic field, and optical boundary measurements are taken as the parameters of the acoustic field vary. In this talk I will give a short introduction to the idea of acousto-optic tomography, and discuss some inverse problems that arise from this imaging technique. In particular, I will describe some recent results for inverse problems derived from radiative transport models of acousto-optic tomography. This is joint work with John Schotland and Guillaume Bal.

We discuss the asymptotic behavior, at a small viscosity, of solutions to some fluid equations related to the Navier-Stokes equations. The model equations, we consider in this talk, are either supplemented with the Navier-slip type boundary conditions, or simplified under some special symmetries or linearized when the no-slip boundary condition is imposed. By explicitly constructing the boundary layer correctors, which approximate the differences between the viscous and inviscid solutions, we validate the smallness of our asymptotic expansions with respect to the viscosity parameter, and prove the vanishing viscosity limit with the optimal rates of convergence.

The motivation for the talk is the recent result, joint with Vincent Guirardel and Camille Horbez, that Out(F_n) admits a topologically amenable action on a Cantor set. This implies the Novikov conjecture for Out(F_n) and its subgroups. Most of the talk will be an introduction to boundary amenability and ways to prove it for simpler groups.

The motivation for the talk is the recent result, joint with Vincent Guirardel and Camille Horbez, that Out(F_n) admits a topologically amenable action on a Cantor set. This implies the Novikov conjecture for Out(F_n) and its subgroups. Most of the talk will be an introduction to boundary amenability and ways to prove it for simpler groups.

This talk will review previous work on quadrupedal gaits and recent work on a generalized model for binocular rivalry proposed by Hugh Wilson. Both applications show how rigid phase-shift synchrony in periodic solutions of coupled systems of differential equations can help understand high level collective behavior in the nervous system.

Probabilistic methods in geometric group theory have recently gained in importance. In the course I will focus on the setting where a countable group G acts on a (possibly nonproper) Gromov hyperbolic space X. Examples include a mapping class group acting on the associated curve complex, or Out(F_n) acting on the complex of free factors, or a group acting on the contact graph of a CAT(0) cube complex it acts on. Under minor assumptions, Maher-Tiozzo show that a random walk on G, projected to X, almost surely converges to a point in the Gromov boundary of X. I will discuss the proof of this theorem. As an application, we will see that "generic" elements of mapping class groups are pseudo-Anosov, and (following Horbez) we will give a random walk proof of the classical theorem of Ivanov classifying subgroups of mapping class groups.

We will give a topological generalization of the planar (p,q) theorem due to Alon and Kleitman. In particular we will show that the assertion of the (p,q) theorem holds for families of open connected sets in the plane under the hypothesis that the intersection of any subfamily is empty or connected. The proof is based on a surprising connection between nerve complexes and complete minors in graphs. This is join work with Minki Kim and Seunghun Lee.

Probabilistic methods in geometric group theory have recently gained in importance. In the course I will focus on the setting where a countable group G acts on a (possibly nonproper) Gromov hyperbolic space X. Examples include a mapping class group acting on the associated curve complex, or Out(F_n) acting on the complex of free factors, or a group acting on the contact graph of a CAT(0) cube complex it acts on. Under minor assumptions, Maher-Tiozzo show that a random walk on G, projected to X, almost surely converges to a point in the Gromov boundary of X. I will discuss the proof of this theorem. As an application, we will see that "generic" elements of mapping class groups are pseudo-Anosov, and (following Horbez) we will give a random walk proof of the classical theorem of Ivanov classifying subgroups of mapping class groups.

Probabilistic methods in geometric group theory have recently gained in importance. In the course I will focus on the setting where a countable group G acts on a (possibly nonproper) Gromov hyperbolic space X. Examples include a mapping class group acting on the associated curve complex, or Out(F_n) acting on the complex of free factors, or a group acting on the contact graph of a CAT(0) cube complex it acts on. Under minor assumptions, Maher-Tiozzo show that a random walk on G, projected to X, almost surely converges to a point in the Gromov boundary of X. I will discuss the proof of this theorem. As an application, we will see that "generic" elements of mapping class groups are pseudo-Anosov, and (following Horbez) we will give a random walk proof of the classical theorem of Ivanov classifying subgroups of mapping class groups.

We consider non-topological solutions of a nonlinear elliptic system problem derived from the SU(3) Chern-Simons models.

The existence of non-topological solutions even for radial symmetric case has been a long standing open problem.

Recently, Choe, Kim, and Lin showed the existence of radial symmetric non-topological solution when the vortex points collapse. However, the arguments in that paper cannot work for an arbitrary configuration of vortex points.

In this talk, I introduce a new approach by using different scalings for different components of the system to construct a family of partial blowing up non-topological solutions.

This talk is based on the joint work with Prof. Chang-Shou Lin and Prof. Ting-Jung Kuo.

In this talk we discuss the generation of interface property for solutions of the nonlocal Allen-Cahn equation which was proposed by Rubinstein and Sternberg as a model for phase separation in a binary mixture. More precisely,

we show that given an arbitrarily initial condition, the solution approaches a step function and hence develops a steep transition layer (interface) within a very short time. Because of the nonlocal term, some PDE tools such as comparison principle cannot be applied so that we have to introduce new method to overcome these diculties. Furthermore, in some cases, we obtain a sharp estimate for the thickness of interface.

This is joint work with Danielle Hilhorst, Hiroshi Matano and Hendrik Weber.

내용: The Atiyah-Singer Index Theorem appeared in 1960’s, which is one of great mathematical achievements in 20th century. It is a far reaching generalization of the Gauss-Bonnet-Chern Theorem for the Euler characteristic, the Hirzebruch Signature Theorem for the Signature of 4k-dimensional compact manifold, the Riemann-Roch-Hirzebruch Theorem for the Arithmetic Genus. Hence to understand this magnificent theorem, we need to investigate how the Euler characteristic, Signature and Arithmetic Genus are expressed by the Fredholm Indices for some appropriate geometric operators. In this talk, I will explain briefly the historical background of the Index Theorem, the Fredholm Indices of elliptic operators and discuss how the Index Theorem was motivated from the above classical celebrated theorems. And then, I will go through very briefly the proof of the Index Theorem by using the heat kernel method. If time permits, I will explain the Index Theorem on a compact manifold with boundary, where the eta-invariant appears as a boundary correction term.

Chudnovsky posed a still open conjectural lower bound for Waldschmidt constants. I will relate this conjecture to certain open conjectures about ideal containments, which themselves are related to Waldschmidt constants.

Let X be a smooth projective rational surface over an algebraically closed field. One version of a long standing conjecture asks whether the set of self-intersections of reduced curves is bounded below. This question can be reduced to studying singular plane curves. One approach is to measure how singular the curve is using a recently introduced quantity called an H-constant. This has raised some new open problems I will discuss.

Waldschmidt constants and Seshadri constants are asymptotic measure of effectivity and nefness related to plane curves. Evaluating them is related to the occurrence of curves of negative self-intersection on blow ups of the plane. I will discuss recent work on computing and bounding these quantities.

The Erdos-Ko-Rado Theorem is a classic result about intersecting families of sets. More recently, analogous “EKR-type” type theorems have been developed for other types of objects. For example, non-trivially intersecting vector spaces, and overlapping strings. In this seminar we will give a proof of the EKR Theorem for permutations in Sn due to Godsil and Meagher. Along the way we will see some useful tools from algebraic graph theory. Namely, a bound on the maximum size of an independent set in a graph, equitable partitions, and eigenpolytopes.

In this talk, I will introduce the distribution of eigenvalues of random normal matrix ensembles. Specifically, I will discuss the existence and universality of scaling limits for the eigenvalues at a bulk singularity, an isolated point in the interior at which the equilibrium density vanishes. I will describe how to find a suitable scale and how the rescaled ward’s identity can be used to prove the universality. This is joint work with Yacin Ameur.

The continuity problem is the question when effective (or Markov computable) maps between effectively given topological spaces are effectively continuous. It will be shown that this is always the case if the the range of the map is effectively bi-regular. As will be shown, such spaces appear quite naturally in the context of the problem.

Evolutionary games of cyclic competitions have been extensively studied to gain insights into one of the most fundamental phenomena in nature:biodiversity that seems to be excluded by the principle of natural selection. The Rock-Paper-Scissors (RPS) game of three species and its extensions [e.g., the Rock-Paper-Scissors-Lizard-Spock (RPSLS) game] are paradigmatic models in this field. In all previous studies, the intrinsic symmetry associated with the cyclic competitions imposes a limitation on the resulting coexistence states, leading to only selective types of such states. We investigate the effect of nonuniform intraspecific competitions on coexistence and find that a wider spectrum of coexistence states can emerge and persist. This surprising finding is substantiated using three classes of cyclic game models through stability analysis, Monte Carlo simulations and patterns of continuous spatiotemporal dynamical evolution. Our finding indicates that intraspecific competitions or alternative symmetry-breaking mechanisms can promote biodiversity to a broader extent than previously thought.

Recent development of high throughput sequencing and CRISPR genome editing technology have brought great advances in the understanding of molecular mechanisms underlying diseases. We studied infertility in Drosophila model: deep sequencing analyses of small RNA, mRNA, and RNA immunoprecipitation sequencing (RIP-seq) showed that a conserved mRNA export factor, Thoc5, represses the expression of transposable elements, which are deleterious mobile genetic elements that cause genome instability and germ cell death. We also found that Thoc5 binds nascent RNAs and facilitates biogenesis of small RNAs, which in turn regulated transposable elements. Furthermore, we used CRISPR-Cpf1, a recently found class 2/type V CRISPR RNA-guided endonuclease that is distinct from the common CRISPR-Cas9, to generate mutant mice. Whole genome analyses of the mutant mice showed precise alteration of DNA sequence at the target genomic locus with no detectable off-target mutation. We generated a cell line with a single base change in Isocitrate Dehydrogenase I (IDH1), a gain-of-function mutation widely found in glioma patients. We further applied CRISPR genome editing to pathogenic bacteria to generate bacterial strains with mutations in genes required for quorum sensing, which is an intercellular communication system for detection cell density. The quorum sensing mutant strains showed reduced biofilm formation, and changes in expression profile of genes in the metabolic pathway.

Among the most well-known examples of L-functions are the Riemann zetafunction and the L-functions associated to classical modular forms.Less well known, but equally important, are the L-functions associatedto Maass forms, which are eigenfunctions of the Laplace-Beltramioperator on a hyperbolic surface. Named after H. Maass, who discoveredsome examples in the 1940s, Maass forms remain largely mysterious.

Fortunately, there are concrete tools to study Maass forms: traceformulas, which relate the spectrum of the Laplace operator on ahyperbolic surface to its geometry. After Selberg introduced hisfamous trace formula in 1956, his ideas were generalised, and varioustrace formulas have been constructed and studied. However, there arefew numerical results from trace formulas, the main obstacle beingtheir complexity. Various types of trace formulas are investigated,constructed and used to understand automorphic representations andtheir L-functions from a theoretical point of view, but most are notexplicit enough to implement in computer code.

Having explicit computations of trace formulas makes many potentialapplications accessible. In this talk, I will explain thecomputational aspects of the Selberg trace formula for GL(2) forgeneral levels and applications towards the Selberg eigenvalueconjecture and classification of 2-dimensional Artin representationsof small conductor.This is a joint work with Andrew Booker and Andreas Strömbergsson.

Imagine you want to present your collection of n coins on a shelf, taking as little space as possible – how should you arrange the coins?

More precisely, we are given n circular disks of different radii, and we want to place them in the plane so that they touch the x-axis from above, such that no two disks overlap. The goal is to minimize the length of the range from the leftmost point on a disk to the rightmost point on a disk.

On this seemingly innocent problem we will meet a wide range of algorithmic concepts: An efficient algorithm for a special case, an NP-hardness proof, an approximation algorithm with a guaranteed approximation factor, APX-hardness, and a quasi-polynomial time approximation scheme.